Term Rewriting System R:
[X, Y]
f(g(X), Y) -> f(X, f(g(X), Y))

Innermost Termination of R to be shown. 



   R
     ↳Dependency Pair Analysis



R contains the following Dependency Pairs:

F(g(X), Y) -> F(X, f(g(X), Y))
F(g(X), Y) -> F(g(X), Y)

Furthermore, R contains one SCC. 


   R
     ↳DPs
       →DP Problem 1
         ↳Rewriting Transformation


Dependency Pairs:

F(g(X), Y) -> F(g(X), Y)
F(g(X), Y) -> F(X, f(g(X), Y))


Rule:


f(g(X), Y) -> f(X, f(g(X), Y))


Strategy:

innermost




On this DP problem, a Rewriting SCC transformation can be performed.
As a result of transforming the rule

F(g(X), Y) -> F(X, f(g(X), Y))
one new Dependency Pair is created:

F(g(X), Y) -> F(X, f(X, f(g(X), Y)))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rewriting Transformation


Dependency Pairs:

F(g(X), Y) -> F(X, f(X, f(g(X), Y)))
F(g(X), Y) -> F(g(X), Y)


Rule:


f(g(X), Y) -> f(X, f(g(X), Y))


Strategy:

innermost




On this DP problem, a Rewriting SCC transformation can be performed.
As a result of transforming the rule

F(g(X), Y) -> F(X, f(X, f(g(X), Y)))
one new Dependency Pair is created:

F(g(X), Y) -> F(X, f(X, f(X, f(g(X), Y))))

The transformation is resulting in one new DP problem: 



   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 3
                 ↳Argument Filtering and Ordering


Dependency Pairs:

F(g(X), Y) -> F(X, f(X, f(X, f(g(X), Y))))
F(g(X), Y) -> F(g(X), Y)


Rule:


f(g(X), Y) -> f(X, f(g(X), Y))


Strategy:

innermost




The following dependency pair can be strictly oriented:

F(g(X), Y) -> F(X, f(X, f(X, f(g(X), Y))))


The following usable rule for innermost w.r.t. to the AFS can be oriented:

f(g(X), Y) -> f(X, f(g(X), Y))


Used ordering: Lexicographic Path Order with Non-Strict Precedence with Quasi Precedence:
trivial

resulting in one new DP problem.
Used Argument Filtering System:
F(x1, x2) -> F(x1, x2)
g(x1) -> g(x1)
f(x1, x2) -> x1


   R
     ↳DPs
       →DP Problem 1
         ↳Rw
           →DP Problem 2
             ↳Rw
             ...
               →DP Problem 4
                 ↳Remaining Obligation(s)




The following remains to be proven:
Dependency Pair:

F(g(X), Y) -> F(g(X), Y)


Rule:


f(g(X), Y) -> f(X, f(g(X), Y))


Strategy:

innermost



Innermost Termination of R could not be shown.
Duration:
0:00 minutes